The most general results on local decay for the Schrödinger evolution
were obtained by Murata [1]}. He derived expansions in time for evolutions \(e^{itH}\)  in all dimensions and with
elliptic \(H=-p(D)+V\)  where \(V\)  is a compact operator in suitable weighted Sobolev spaces. As a general rule, the coefficients
in these expansions corresponding to nongeneric threshold behavior (i.e., slow decay resulting from threshold eigenvalues or resonances)
are finite rank operators which can be computed in terms of the eigenfunctions and resonant states. As an example, the
one-dimensional free evolution satisfies
\(e^{-it\partial _x^2} f(x) = ct^{-\frac{1}{2}} \int f(y)\, dy + \rho (t) f(x),\qquad \Vert  \langle \cdot \rangle ^{-\sigma } \rho (t)f\Vert _2\lesssim t^{-\frac{3}{2}}\Vert \langle \cdot \rangle ^\sigma f\Vert _2\) 
